If you cut a triangle out of a piece of paper and put your pencil point at the centroid, you would be able to balance the triangle there.) (This point is the center of mass for the triangle. The medians are concurrent they meet at a point called the centroid of the triangle.The following surprising facts are true for every triangle: When three or more lines meet at a single point, they are said to be concurrent. Problems C3 and C4 and the Video Segment problems taken from Connected Geometry, developed by Educational Development Center, Inc. You can find this segment on the session video approximately 15 minutes and 40 seconds after the Annenberg Media logo.
What are the similarities and differences in your results? What conjectures can you make about the constructions you’ve just completed? Compare your solutions to Problem C4 with those in this video segment. In this video segment, participants construct the altitudes, medians, and midlines of their triangles. Carefully construct the three angle bisectors of the fifth triangle.Carefully construct the three perpendicular bisectors of the fourth triangle.Carefully construct the three midlines of the third triangle.Carefully construct the three medians of the second triangle.Carefully construct the three altitudes of the first triangle.Use one triangle for each construction below. And remember that the altitude may fall outside the triangle, so you might want to draw or fold an extension of the sides of the triangle to help you.ĭraw five triangles, each on its own piece of patty paper. When you construct altitudes, you need to construct a perpendicular to a segment, but not necessarily at the midpoint of that segment. Except in the case of special triangles (such as an equilateral triangle, and one median in an isosceles triangle), you can’t construct a median with just one fold. When you construct medians, you need to do two things: First find the midpoint then fold or draw a segment connecting that point to the opposite vertex. The intersection of the crease and the original line segment is the midpoint of the line segment.
Next, fold the paper so that the endpoints of the line segment overlap.
To construct the midpoint of a line segment, start by drawing a line segment on the patty paper. Here is a sample construction with patty paper to get you started: Throughout this part of the session, use just a pen or pencil, your straightedge, and patty paper to complete the constructions described in the problems. Though your “straightedge” might actually be a ruler, don’t measure! Use it only to draw straight segments. Since you can see through the paper, you can use the folds to create geometric objects. You can fold the patty paper to create creases. In the problems below, your tools will be a straightedge and patty paper. The most common tools for constructions in geometry are a straightedge (a ruler without any markings on it) and a compass (used for drawing circles). A construction is a method, while a picture merely illustrates the method. It shows how a figure can be accurately drawn with a specified set of tools. The essential element of a construction is that it is a kind of guaranteed recipe. Drawings are intended to aid memory, thinking, or communication, and they needn’t be much more than rough sketches to serve this purpose quite well. Geometers distinguish between a drawing and a construction.